Nicolas Gastineau

University of Burgundy | UB · Laboratoire Électronique, Informatique et Image (LE2I)

The study of a variation of the marking game, in which the first player marks vertices and the second player marks edges of an undirected graph was proposed by Bartnicki et al. (Electron J Combin 15:R72, 2008). In this game, the goal of the second player is to mark as many edges around an unmarked vertex as possible, while the first player wants ju...

In this paper, we present two deterministic leader election algorithms for programmable matter on the face-centered cubic grid. The face-centered cubic grid is a 3-dimensional 12-regular infinite grid that represents an optimal way to pack spheres (i.e., spherical particles or modules in the context of the programmable matter) in the 3-dimensional...

Given a graph G and a nondecreasing sequence \(S=(s_1,\ldots ,s_k)\) of positive integers, the mapping \(c:V(G)\longrightarrow \{1,\ldots ,k\}\) is called an S-packing coloring of G if for any two distinct vertices x and y in \(c^{-1}(i)\), the distance between x and y is greater than \(s_i\). The smallest integer k such that there exists a \((1,2,...

In this article, we present two deterministic leader election algorithms for programmable matter on the face‐centered cubic grid. The face‐centered cubic grid is a three‐dimensional 12‐regular infinite grid that represents an optimal way to pack spheres (i.e., spherical particles or modules in the context of the programmable matter) in the three‐di...

Given a graph G, the exact distance-p graph \(G^{[\natural p]}\) has V(G) as its vertex set, and two vertices are adjacent whenever the distance between them in G equals p. We present formulas describing the structure of exact distance-p graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distanc...

The context of this paper is programmable matter, which consists of a set of computational elements, called particles, in an infinite graph. The considered infinite graphs are the square, triangular and king grids. Each particle occupies one vertex, can communicate with the adjacent particles, has the same clockwise direction and knows the local po...

The face-centered cubic grid is a three dimensional 12-regular infinite grid. This graph represents an optimal way to pack spheres in the three-dimensional space. We give lower and upper bounds on the chromatic number of the dth power of the face-centered cubic grid. In particular, in the case d = 2 we prove that the chromatic number of this grid i...

Given a graph $G$, the exact distance-$p$ graph $G^{[\natural p]}$ has $V(G)$ as its vertex set, and two vertices are adjacent whenever the distance between them in $G$ equals $p$. We present formulas describing the structure of exact distance-$p$ graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exa...

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $...

The context of this paper is programmable matter, which consists of a set of computational elements, called particles, in an infinite graph. The considered infinite graphs are the square, triangular and king grids. Each particle occupies one vertex, can communicate with the adjacent particles, has the same clockwise direction and knows the local po...

The context of this paper is programmable matter, which consists of a set of computational elements, called particles, in an infinite graph. The considered infinite graphs are the square, triangular and king grids. Each particle occupies one vertex, can communicate with the adjacent particles, has the same clockwise direction and knows the local po...

The face-centered cubic grid is a three dimensional 12-regular infinite grid. This graph represents an optimal way to pack spheres in the three-dimensional space. In this grid, the vertices represent the spheres and the edges represent the contact between spheres. We give lower and upper bounds on the chromatic number of the d th power of the face-...

Given a non-decreasing sequence S = (s 1,s 2,. .. ,s k) of positive integers, an S-packing edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 ,X 2,. .. ,X k } such that for each 1 $\le$ i $\le$ k, the distance between two distinct edges e, e ' $\in$ X i is at least s i + 1. This paper studies S-packing edge-colorings...

The question of whether subcubic graphs have finite packing chromatic number or not is still open although positive responses are known for some subclasses, including subcubic trees, base-3 Sierpiski graphs and hexagonal lattices. In this paper, we answer positively to the question for some subcubic outerplanar graphs. We provide asymptotic bounds...

The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by defining (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges,...

Influence in Twitter has become recently a hot research topic, since this micro-blogging service is widely used to share and disseminate information. Some users are more able than others to influence and persuade peers. Thus, studying most influential users leads to reach a large-scale information diffusion area, something very useful in marketing...

Gyárfás et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\Gamma(G)\ge t$ if and only if $G$ contains an induced subgraph called a $t$-atom.The family of $t$-atoms has bounded order and contains a finite number of graphs.In this article, we introduce equivalents of $t$-atoms for b-coloring and partial Grundy coloring.Thi...

This work establishes the complexity class of several instances of the S-packing coloring problem: for a graph G, a positive integer k and a non decreasing list of integers S = (s_1 , ..., s_k ), G is S-colorable, if its vertices can be partitioned into sets S_i , i = 1,... , k, where each S_i being a s_i -packing (a set of vertices at pairwise dis...

Dans un réseau, la recherche de plusieurs arbres couvrants avec des propriétés intéressantes a amené à l’introduction de plusieurs notions : les arbres couvrants arête-disjoints, les arbres indépendants enracinés en un sommet et les arbres complètement indépendants. Afin de généraliser ces notions, nous introduisons la notion d’arbres couvrants (i,...

An $i$-packing in a graph $G$ is a set of vertices at pairwise distance greater than $i$. For a nondecreasing sequence of integers $S=(s\_{1},s\_{2},\ldots)$, the $S$-packing chromatic number of a graph $G$ is the least integer $k$ such that there exists a coloring of $G$ into $k$ colors where each set of vertices colored $i$, $i=1,\ldots, k$, is a...

Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\Gamma(G)\ge t$ if and only if $G$ contains an induced subgraph called a $t$-atom. The family of $t$-atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of $t$-atoms for b-coloring and partial Grundy co...

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices, except the vertices $u$ and $v$, then $T_1,\ldots, T_k$ are completely independent spanning trees in $G$. For $2k$-...

The Grundy number of a graph G, denoted by Gamma(G), is the largest k such that there exists a partition of V(G), into k independent sets V-1, . . . , V-k and every vertex of V-i is adjacent to at least one vertex in V-j, for every j < i. The objects which are studied in this article are families of r-regular graphs such that Gamma(G) = r + 1. Usin...

Our research are about graph coloring with distance constraints (packing coloring) or neighborhood constraints (Grundy coloring). Let S={si| i in N*} be a non decreasing sequence of integers. An S-packing coloring is a proper coloring such that every set of color i is an si-packing (a set of vertices at pairwise distance greater than si). A graph G...

Nous étudions l’existence de r arbres couvrants complètement disjoints dans des graphes 2r-réguliers et 2r-connexes, et énonçons des conditions nécessaires à leur existence. Nous déterminons le nombre maximum d’arbres dans les produits cartésiens d’une clique et d’un cycle. Nous montrons que ce nombre n’est pas toujours r.

Given a non-decreasing sequence $S=(s_1,s_2, \ldots, s_k)$ of positive integers, an {\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\{s_1,s_2, \ldots, s_k\}$ such that any two vertices with color $s_i$ are at mutual distance greater than $s_i$, $1\le i\le k$. This paper studies $S$-packing colorings of (sub)cubic graphs....

Given a non-decreasing sequence $S=(s_1,s_2, \ldots, s_k)$ of positive integers, an {\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\{s_1,s_2, \ldots, s_k\}$ such that any two vertices with color $s_i$ are at mutual distance greater than $s_i$, $1\le i\le k$. This paper studies $S$-packing colorings of (sub)cubic graphs....

This work establishes the complexity class of several instances of the S-coloring problem: For a graph G, a positive integer k and a non decreasing list of integers S = (s1, . . ., sk), G admits a S-coloring, if its vertices can be partitioned into sets Xsi, i = 1, . . ., k, where each Xsi being an si -packing (a set of vertices at pairwise distanc...